The answer is 1600. Easy as pie.
he likes 25 because he can get square root (5), but he cannot get square root from 24 (actually, he can, but it will be 2*((6)^-1) so...), he can get also from 400 (20), but not from 300, and from 144 (12), but not from 145. From above answers only from 1600 we can get square root which result is a natural number (40).